>>12877348For picture
>>12877366You have a function and we want to find the area under its curve.
The Riemann sums idea is to use rectangles to approximate the area under the curve of the function. For example, if your curve went from 0 to 3, you could fit 3 rectangles of length 1 and each with a certain height. Let's say the height is such that your rectangles are above the function always. Look at the upper sum image, it would be like that.
Now, if instead of 3 rectangles, we used 4. Then, the amount of area within the rectangles, but above the curve (the green part) would be less than if we had used just 3 rectangles. This means our approximation would be better, because the green area is area that is in excess of the real curve's area.
If we repeat this process with 5, 6, 7, ..., n rectangles we would get the best approximation of an upper sum possible as n goes towards infinity (like a really large number). This means we approximate using infinitely many rectangles of very small length (the length here being delta x, a small variation of x).
We can repeat this process, but by having rectangles underneath the curve strictly, where, like in the lower sum image, the pink area would be the area we failed to calculate.
As we have infinitely many rectangles, our approximation would get better.
Now, we have two approximations. The upper sum which has an excess area (the green part) and the lower sum which missed some area (the pink part). This means the actual area of the curve is between the lower sum (lower bound) and the upper sum (upper bound).
We notice that as our number of rectangles n goes towards infinity, that both the upper sum and lower sum give the name value for the Area.
This means that the area underneath the curve that we will denote A is bounded strictly by:
Lower Sum <= A <= Upper Sum.
However, the Lower Sum = Upper Sum and let's say their value is S.
So S <= A <= S. This implies that A = S. We found our area. This is Riemann sums.
